This is pretty cute! I like the character!

I feel like I'm just posting a lot of videos on here recently, but this one explains a run cycle that I think you could adapt.
https://www.youtube.com/watch?v=_jt_tbF4o4MIt should help with leg positions, and when to raise and lower the character. Also, the hair waving is equivalent to your character's hat.
For specific pixel-art tips, I can share some ideas.
Some of the angles you're using for the legs look a bit odd to me. Frames 5 and 10 especially. Creating an animation by moving one row of pixels in a column, then rippling that effect up doesn't really make me think that column is rotating.
Instead, if you can, try to use lots of 45º angles and straight lines, instead of shallower angles. I think they tend to look better in pixel art, as they don't create 'jaggies'. And I would try to think in terms of where joints might be, so I'd bend only at the "knee". You only have a few pixels to work with, I know, but that also means that a pixel out of place is more obvious.
I find frames 1 and 6 are also a bit odd, with the legs fully extended backwards and forwards and also both on the ground. I feel like the rear leg needs to be off the ground and pointing backwards instead of down. But the video should cover this.
I'm not seeing any arm movement. The character kinda looks like a burglar with sack of swag, so it could be holding it closed. Is there some story here as to why the arms are stationary?
Finally I guess this is technically a walk cycle. Running is defined by there being times when both feet are off the ground, but here they are always in contact. It's just semantics, but on the other hand, if you want to get across the idea of a bit more energy, maybe include some "air" frames? Again I believe the video covers this, but if you don't like the run cycle in there, there are plenty around for you to get inspiration from. I quite the Preston Blair one, but put "Run cycle animation" into Google Images and see if there are any you like.
I hope there's something in there that's helpful.